Meaning of representations of groups in different dimensions

[Screwdriver]

Problem

This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions.

Relevant Example

Take $SO(3)$ for example; it's the group of $3\times 3$ orthogonal matrices of determinant $1$ under matrix multiplication. We can represent elements of the group as $3 \times 3$ rotation matrices, so it makes sense to interpret the group itself as a rotation group in $\mathbb{R}^3$. The rotation preserves the volume and radii of a collection of points. What I don't understand is the meaning of a representation of this group in terms of matrices of dimension other than $3$. The definition of the group itself seems to assume $3 \times 3$ matrices, so how do matrices of higher dimension make sense, and what is their interpretation? Are they still rotations in $\mathbb{R}^3$ ? What are the preserved quantities?

What I Know

I get that $SO(3)$ has three generators because there are three continuous parameters needed to define the group, and that as long as they satisfy the commutation relations you can have any representation you want (so, matrices of a higher dimension are mathematically allowed) but I can't figure out what the other representations mean. I also understand that, for example, different representations of $SU(2)$ correspond to different spins, but that just seems more abstract.

 

[Orodruin]

A representation is just a map from the group to a set of matrices which preserves the group structure. It does not need to have an inverse.

For example, you can represent any group with the trivial mapping $g\to 1$ (this preserves the group structure of any group). If you take this representation, quantities that transform according to it would be scalar products of vectors (or any scalar, such as the volume spanned by a set of vectors). Things that transform under the fundamental representation are vectors (the vector components are not the same after rotation).

 

[Screwdriver]

A representation is just a map from the group to a set of matrices which preserves the group structure. It does not need to have an inverse.

For example, you can represent any group with the trivial mapping $g\to 1$ (this preserves the group structure of any group). If you take this representation, quantities that transform according to it would be scalar products of vectors (or any scalar, such as the volume spanned by a set of vectors). Things that transform under the fundamental representation are vectors (the vector components are not the same after rotation).

One $3$-dimensional representation of $SO(3)$ is $3\times 3$ rotation matrices parametrized by three Euler angles. The representation acts on $3$-dimensional vectors and rotates them in $\mathbb{R}^3$. That makes sense. What doesn't make sense is the interpretation of, say, the $4$-dimensional representation. A $4$-dimensional representation of $SO(3)$ is $4 \times 4$ matrices which act on $4$-dimensional vectors. Evidently these are not rotations in $\mathbb{R}^4$, because rotations in $\mathbb{R}^4$ are parametrized by $6$ Euler angles. So what are they?

Wait a minute. Is it that the $4$-dimensional representation of $SO(3)$ rotates vectors in $\mathbb{R}^4$ through a $3$-dimensional subspace of $\mathbb{R}^4$? It's hard to think in $4$ dimensions, but would the lower-dimensional analogy be that the $3$-dimensional representation of $SO(2)$ rotates vectors in $\mathbb{R}^3$ around (say) the $z$ axis, whereas the $3$-dimensional representation of $SO(3)$ rotates vectors in $\mathbb{R}^{3}$ rotates vectors around any axis?

 

[Orodruin]

Since the irreps of SU(3) are odd-dimensional (there is a one-to-one correspondence to spherical harmonics), yes, your four dimensional representation must be either 3+1 or 1+1+1+1 when broken down into irreps. This means that it is either a vector + a scalar or four scalars.

However, you can think of representations of SO(3) which are higher dimensional but still irreducible, i.e., all spherical harmonics with l > 1.

 

[Screwdriver]

However, you can think of representations of SO(3) which are higher dimensional but still irreducible, i.e., all spherical harmonics with l > 1.

So what do those have to do with rotations in $\mathbb{R}^3$?

 

[Orodruin]

They are representations of $\mathbb R^3$. In the case of $\ell = 2$, it is a five-dimensional functional space which is spanned by five functions which rotate into one another under the action of the SO(3) representation. You cannot reduce this representation to the action on a vector or a scalar.